Zero measure spectrum for multi-frequency Schrödinger operators

Limit-periodic Continuum Schrödinger Operators with Zero Measure Cantor Spectrum

We consider Schrödinger operators on the real line with limitperiodic potentials and show that, generically, the spectrum is a Cantor set of zero Lebesgue measure and all spectral measures are purely singular continuous. Moreover, we show that for a dense set of limit-periodic potentials, the spectrum of the associated Schrödinger operator has Hausdorff dimension zero. In both results one can i...

Locating the Spectrum for Magnetic Dirac and Schrödinger Operators

Singular Spectrum of Lebesgue Measure Zero for One-dimensional Quasicrystals

The spectrum of one-dimensional discrete Schrödinger operators associated to strictly ergodic dynamical systems is shown to coincide with the set of zeros of the Lyapunov exponent if and only if the Lyapunov exponent exists uniformly. This is used to obtain Cantor spectrum of zero Lebesgue measure for all aperiodic subshifts with uniform positive weights. This covers, in particular, all aperiod...

Positive Measure Spectrum for Schrödinger Operators with Periodic Magnetic Fields

We study Schrödinger operators with periodic magnetic field in R2, in the case of irrational magnetic flux. Positive measure Cantor spectrum is generically expected in the presence of an electric potential. We show that, even without electric potential, the spectrum has positive measure if the magnetic field is a perturbation of a constant one. Introduction Magnetic Schrödinger operators have b...

Measure of non strict singularity of Schechter essential spectrum of two bounded operators and application

In this paper‎, ‎we discuss the essential spectrum of sum of two bounded operators‎ ‎using measure of non strict singularity‎. ‎Based on this new investigation‎, ‎a problem of one-speed neutron transport operator is presented‎.